A computational method for solution of the prey and ...

Applied Mathematics and Computation 163 (2005) 841–847 www.elsevier.com/locate/amc

A computational method for solution of the prey and predator problem J. Biazar a

a,*

, R. Montazeri

b

Department of Mathematics, Faculty of science, Guilan University, P.O. Box 1914, Rasht, P.C. 41938, Iran b Islamic Azad University, Branch of Lahijan, P.O. Box 1616, Lahijan, Iran

Abstract In this article a mathematical model of the problem of prey and predator being presented and Adomian decomposition method is employed to compute an approximation to the solution of the system of nonlinear Volterra differential equations governing on the problem. Some plots for the population of the prey and predator versus time are presented to illustrate the solution. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Adomian decomposition method; System of nonlinear differential equations

1. Introduction There are some rabbits and foxes living together. Foxes eat the rabbits and rabbits eat clover. Suppose that there are enough clovers and the rabbits have enough food to eat. When there are a lot of rabbits, the foxes also grow and their population increase. When the number of foxes increase and they eat a lot of rabbits they enter into a short period of food and their number decrease. As the number of the foxes decrease, the rabbits will be safe and their population increase. When the number of rabbits increase the number of foxes would increase and by passing the time we can see an infinite repeatability of increase and decrease in the population of these two kinds of animals (Figs. 1–4).

*

Corresponding author. E-mail addresses: [email protected], [email protected] (J. Biazar).

0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.05.001

842

J. Biazar, R. Montazeri / Appl. Math. Comput. 163 (2005) 841–847

Fig. 1. Numbers of rabbits and faxes versus time.

Fig. 2. Numbers of rabbits and faxes versus time.

The governing equations to the problem would be as follows [1]. 8 dx > < ¼ xðtÞða  byðtÞÞ; dt dy > : ¼ yðtÞðc  dxðtÞÞ; dt

ð1Þ

where xðtÞ and yðtÞ are respectively the populations of rabbits and the foxes at the time t.

J. Biazar, R. Montazeri / Appl. Math. Comput. 163 (2005) 841–847

843

Fig. 3. Numbers of rabbits and faxes versus time.

Fig. 4. Numbers of rabbits and faxes versus time.

2. Computational method for solving (1) To solve the system of Eqs. (1) Adomian decomposition method, well addressed in [2,3], is employed. The equivalent canonical form of this system is as follows: ( Rt xðtÞ ¼ xðt ¼ 0Þ þ 0 xða  byÞ dt; Rt ð2Þ yðtÞ ¼ yðt ¼ 0Þ  0 yðc  dxÞ dt:

844

J. Biazar, R. Montazeri / Appl. Math. Comput. 163 (2005) 841–847

As usual in Adomian decomposition method the solutions of Eqs. (2) are considered to be as the sum of the following series: x¼

1 X

xn ;

y¼

n¼0

1 X

ð3Þ

yn :

n¼0

And the integrand in Eq. (2), as the sum of the following series. f ðx; yÞ ¼

1 X

An ðx0 ; . . . ; xn ; y0 ; . . . ; yn Þ

ð4Þ

Bn ðx0 ; . . . ; xn ; y0 ; . . . ; yn Þ

ð5Þ

n¼0

gðx; yÞ ¼

1 X n¼0

where An , Bn are called Adomian polynomials [2]. Substituting (3)–(5) into (2) we get: Z tX 1 1 X xn ¼ xðt ¼ 0Þ þ An ðx0 ; . . . ; xn ; y0 ; . . . ; yn Þ dt n¼0

¼ xðt ¼ 0Þ þ

0

n¼0 t

n¼0

0

1 Z X

An ðx0 ; . . . ; xn ; y0 ; . . . ; yn Þ dt

ð6Þ

and 1 X

yn ¼ yðt ¼ 0Þ þ

Z

t 0

n¼0

¼ yðt ¼ 0Þ þ

1 X n¼0

1 X n¼0 Z t

Bn ðx0 ; . . . ; xn ; y0 ; . . . ; yn Þ dt Bn ðx0 ; . . . ; xn ; y0 ; . . . ; yn Þ dt:

ð7Þ

0

From which we define the following scheme: x0 ¼ xðt ¼ 0Þ; y0 ¼ yðt ¼ 0Þ; Rt xnþ1 ¼ 0 An ðx0 ; . . . ; xn ; y0 ; . . . ; yn Þ dt; Rt ynþ1 ¼ 0 Bn ðx0 ; . . . ; xn ; y0 ; . . . ; yn Þ dt;

n ¼ 0; 1; 2; . . .

ð8Þ

Computing Adomian polynomials by the algorithm presented in [4], yields to A0 ðx0 ; y0 Þ ¼ ax0  bx0 y0 A1 ðx0 ; x1 ; y0 ; y1 Þ ¼ ax1  bðx0 y1 þ x1 y0 Þ A2 ðx0 ; x1 ; x2 ; y0 ; y1 ; y2 Þ ¼ ax2  bðx0 y2 þ x1 y1 þ x2 y0 Þ A3 ðx0 ; x1 ; x2 ; x3 ; y0 ; y1 ; y2 ; y3 Þ ¼ ax3  bðx0 y3 þ x1 y2 þ x2 y1 þ x3 y0 Þ .. .

ð9Þ

J. Biazar, R. Montazeri / Appl. Math. Comput. 163 (2005) 841–847

845

and B0 ðx0 ; y0 Þ ¼ cy0  dx0 y0 B1 ðx0 ; x1 ; y0 ; y1 Þ ¼ cy1  dðx1 y0 þ x0 y1 Þ B2 ðx0 ; x1 ; x2 ; y0 ; y1 ; y2 Þ ¼ cy2  dðx2 y0 þ x1 y1 þ x0 y2 Þ B3 ðx0 ; x1 ; x2 ; x3 ; y0 ; y1 ; y2 ; y3 Þ ¼ cy3  dðx3 y0 þ x2 y1 þ x1 y2 þ x0 y3 Þ .. . Substituting in (9) we would have: Z x x1 ¼ A0 ðx0 ;y0 Þdt ¼ ax0 t  bx0 y0 Z0 t y1 ¼ B0 ðx0 ;y0 Þdt ¼ cy0 t  dx0 y0 t Z0 t x2 ¼ Aðx0 ;x1 ;y0 ;y1 Þdt ¼ 12ðbx0 ðcy0 t þ dx0 y0 tÞ  bðax0 t  bx0 y0 tÞy0 0

þ aðax0 t  b0 y0 tÞÞt2 Z t y2 ¼ B1 ðy0 y1 Þdt ¼ 12ðdy0 ðax0 t þ bx0 y0 tÞ  dðcy0 t þ dx0 y0 tÞx0 0

þ cðcy0 t þ dx0 y0 tÞÞt2 Z t x3 ¼ A2 ðx0 ;x1 ;x2 ;y1 ;y2 ;y3 Þdt ¼ 13ð12bx0 ðdy0 ðax0  bx0 y0 Þ  dðcy0 þ dx0 y0 Þx0 0

þ cðcy0 þ dx0 y0 ÞÞt2  bðax0 t  bx0 y0 tÞðcy0 t þ dx0 y0 tÞ  12bðbx0 ðcy0 þ dx0 y0 Þ  bðax0  bx0 y0 Þy0 þ aðax0  bx0 y0 ÞÞt2 y0 þ 12aðbx0 ðcy0 þ dx0 y0 Þ  bðax0  bx0 y0 Þy0 þ aðax0  bx0 y0 ÞÞt3 Z t y3 ¼ B2 ðx0 ;x1 ;x2 ;y1 ;y2 ;y2 Þdt ¼ 13ð12dy0 ðbx0 ðcy0 þ bx0 y0 Þ  bðax0  bx0 y0 Þy0 0

þ aðax0  bx0 y0 ÞÞt2  bðax0 t  bx0 y0 tÞðcy0 t þ dx0 y0 tÞ  12bðbx0 ðcy0 þ dx0 y0 Þ  dðcy0 þ dx0 y0 Þx0 þ cðcy0 þ dx0 y0 ÞÞt2 x0  12cðdy0 ðax0  bx0 y0 Þ  dðcy0 þ dx0 y0 Þy0 þ cðcy0 þ dx0 y0 ÞÞt3 .. .

A four terms approximation to the solutions are considered; xðtÞ  x0 þ x1 þ x2 þ x3 ; yðtÞ  y0 þ y1 þ y2 þ y3 : For the convergence of the method the reader is referred to [5].

846

J. Biazar, R. Montazeri / Appl. Math. Comput. 163 (2005) 841–847

3. Numerical results and discussion For numerical study the following values are used. Case

x0

y0

a

b

c

d

1 2 3 4

14 14 16 16

18 18 10 10

1 0.1 0.1 1

1 1 1 1

0.1 1 1 0.1

1 1 1 1

According to the values introduced in the table the following solutions are derived. Case 1 : Case 2 : Case 3 : Case 4 :

xðtÞ  14  238t  271:6t2 þ 20191:31333t3 ; yðtÞ  18 þ 250:2t  197:9t2  8794:533333t3 ; xðtÞ  14  250:6t þ 604:87t2 þ 19364:94233t3 ; yðtÞ  18 þ 234t  734:4t2  19546:40333t3 ; xðtÞ ¼ 16  158:4t  415:92t2 þ 7516:536t3 ; yðtÞ ¼ 10 þ 150t þ 333t2  7391:76t3 ; xðtÞ ¼ 16  144t  624t2 þ 6602:4t3 ; yðtÞ ¼ 10 þ 159t þ 544:05t2  6828:53t3 :

The following plots show the relations between the number of foxes and the rabbits versus time. As the plots state the number of foxes increase, as the number of rabbits, the source of food for foxes, decrease. Foxes will reach their maximum as the rabbits reach their minimum. Since the number of rabbits has decreased there are not enough food for foxes, so their number would decrease, and so on. Maple 9 is used to carry computations. The analytical approximations to the solutions are reliable, and confirms the power and ability of the Adomian decomposition method as an easy device for computing the solution of a nonlinear system of differential equations. References [1] G.F. Simmons, Differential Equations with Applications and Historical Notes, McGraw-Hill, 1972. [2] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordecht, 1994. [3] G. Adomian, G.E. Adomian, A global method for solution of complex systems, Math. Model 5 (1984) 521–568.

J. Biazar, R. Montazeri / Appl. Math. Comput. 163 (2005) 841–847

847

[4] J. Biazar, E. Babolian, G. Kember, A. Nouri, R. Islam, An alternate algorithm for computing Adomian polynomials, Appl. Math. Comput. 38 (2–3) (2003) 523–529. [5] J. Biazar, E. Babolian, R. Islam, Solution of systems of ordinary differential equations with Adomian decomposition method, Appl. Math. Comput. 147 (3) (2004) 713–719.